For time-harmonic electromagnetic waves with exp (-i.omega.t) excitation, chirality is introduced into electromagnetic theory through the constitutive relations given by EQU D=.epsilon.E+i.xi..sub.c B (1) EQU H=B/.mu.+i.xi..sub.c E (2)
for the lossless case where boldface quantities denote vectors and plain text denotes scalars. The "chirality admittance" .xi..sub.c (a real number) is an indication of the degree of chirality of the medium and .epsilon. and .mu. (real numbers) are the usual permittivity and permeability, respectively. For lossy chiral media, the quantities .epsilon., .mu. and .xi..sub.c become complex numbers.
A "chiral medium" comprises chiral objects of the same handedness, randomly oriented and uniformly distributed in a host or background material. This host or background material may comprise common dielectric and/or magnetic materials wherein the chiral objects are embedded. A chiral object is a three-dimensional body that cannot be brought into congruence with its mirror image by translation and rotation. Therefore, all chiral objects can be classified in terms of their "handedness." The term "handedness," as known by those with skill in the art, refers to whether a chiral object is "right-handed" or "left-handed." That is, if a chiral object is right-handed (left-handed), its mirror image is left-handed (right-handed). Therefore, the mirror image of a chiral object is its enantiomorph and is said to be of opposite handedness.
Chiral media exhibit electromagnetic chirality which embraces optical activity and circular dichroism. Optical activity refers to the rotation of the plane of polarization of optical waves by a medium while circular dichroism indicates a change in the polarization ellipticity of optical waves by a medium. (There exists a variety of materials that exhibit optical activity. For example, for 0.63-mm wavelength, TeO.sub.2 exhibits optical activity with a chirality admittance magnitude of 3.83.times.10.sup.-7 mho. This results in a rotation of the plane of polarization of 87.degree. per mm. Similar phenomena exist at microwave and millimeter wave frequencies when chiral objects of like handedness are embedded in a background or host medium.) These phenomena, known since the mid nineteenth century, are due to the presence of the two unequal characteristic wavenumbers corresponding to two circularly polarized eigenmodes with opposite handedness. The fundamentals of electromagnetic chirality are known. See, e.g., J. A. Kong, Theory of Electromagnetic Waves, pages 2-8, 77-79 (1975); E. J. Post, Formal Structure of Electromagnetics, pages 127-137, 171-176 (1962). More recent work includes the macroscopic treatment of electromagnetic waves with chiral structures, D. L. Jaggard et al., Applied Physics "On Electromagnetic Waves in Chiral Media," 18, 211-216, (1979); the analysis of dyadic Green's functions and dipole radiation in chiral media, S. Bassiri et al., Alta Frequenza "Dyadic Green's Function and Dipole Radiation in Chiral Media," 2, 83-88, (1986) and N. Engheta et al., IEEE Trans. on Ant. & Propag. "One- and Two-Dimensional Dyadic Green's Functions in Chiral Media," 37, 4, (1989); and the reflection and refraction of waves at a dielectric-chiral interface, S. Bassiri et al., J. Opt. Soc. Am. A5, "Electromagnetic Wave Propagation Through a Dielectric-Chiral Interface and Through a Chiral Slab," 1450-1459, (1988).
Using the time-harmonic Maxwell equations for both electric sources J and .rho. and magnetic sources J.sub.m and .rho..sub.m yields ##EQU1## From these relations, the following inhomogeneous differential equations for the field quantities can be found with the aid of (1) and (2). ##EQU2## where the chiral differential operator is defined by the relation EQU .quadrature..sub.c.sup.2 { }.ident..gradient..times..gradient..times.{ }-2.omega..mu..xi..sub.c .gradient..times.{ }-k.sup.2 { } (11)
and where ##EQU3## is a generalized "chiral impedance" with .eta..sub.o (=.sqroot..mu./.epsilon.) as the background intrinsic wave impedance. The introduction of both the chiral impedance by relation (12) and the chiral admittance through expressions (1)-(2) leads to the definition of a dimensionless "chirality factor" .kappa. given by the product: EQU .kappa..ident..eta..sub.c .xi..sub.c, (13)
where the absolute value of .kappa. is bounded by zero and unity. This parameter is a quantitative measure of the degree of chirality of the medium.
Since the fields E, B, D and H are linearly dependent on the current sources J and J.sub.m one can write these fields in terms of integrals over the sources and an appropriately defined dyadic Green's function. Furthermore, these expressions can be simplified so that each field eigenmode, denoted by a ".+-." subscript is written in the form below: ##EQU4## where the dyadic Green's function .GAMMA.(x,x') is given below and x is the observation vector (x,y,z) and x' is the source vector (x',y',z'). Here boldface quantities denote vectors while underbars indicate dyads. It is noted that the total field quantities are the sum of the "+" and the "-" eigenmodes given in (14)-(17). Each eigenmode represents a circularly polarized wave of a given handedness.
The dyadic Green's function .GAMMA.(x,x') can be rewritten in the compact form EQU .GAMMA.(x,x')=.GAMMA..sup.+ (x,x')+.GAMMA..sup.- (x,x')=.beta..gamma..sup.+ EQU (k.sub.+)G.sub.+ (x,x')+[1-.beta.].gamma..sup.- (k.sub.--)G.sub.-- (x,x')(18)
where the "+" and "-" superscripts refer to the first and second terms, respectively, on the right-hand side of (18) and the dyadic operators for the two eigenmodes are given in terms of the unit dyad I by EQU .gamma..sup..+-. (k.sub..+-.)={I.+-.k.sub..+-..sup.-1 I.times..gradient.+k.sub..+-..sup.-2 .gradient..gradient.}(19)
and where ##EQU5## The wavenumbers k.sub..+-. are the propagation constants for the two eigenmodes ("+" and "-") supported by the medium. The factors .beta. and 1-.beta. are denoted "handedness factors". These quantities will play a role in the far-field radiation patterns of antennas and arrays and represent the relative amplitude of waves of each handedness. Here k.sub.o (=.omega..sqroot..mu..epsilon.) is the host or background wavenumber of the achiral (meaning not chiral) media with identical permittivity and permeability.
From a far-field expansion of the Green's dyad (18) the electric field eigenmodes corresponding to (14) can be written in the form ##EQU6## for general current sources where r=.vertline.x.vertline., e.sub.r is a unit vector along the observation position vector x. It is understood here and in the following equations that in the triple cross product involving e.sub.r the cross products are carried out right to left. Likewise, using (15) it can be shown that the magnetic field in this limit is given by the relation ##EQU7##
Of particular note from (24)-(25) is that either eigenmode can be excited while the other is suppressed through the appropriate choice of electric and magnetic sources.